Problem:
Rain is falling vertically at a speed of 20m/s. a person is running in the rain with a velocity of 5 m/s and a wind also starts blowing at 15 m/s(both from the west). the angle with the vertical which the person should hold his umbrella so that he may not get drenched is?

Solution:
To solve this problem, we need to calculate the velocity of the rain relative to the person. We can do this by using vector addition.

Step 1: Calculate the velocity of the rain relative to the ground.
The velocity of the rain relative to the ground is given as 20 m/s vertically downwards.

Step 2: Calculate the velocity of the person relative to the ground.
The velocity of the person relative to the ground is given as 5 m/s towards the east.

Step 3: Calculate the velocity of the rain relative to the person.
The velocity of the rain relative to the person is the vector difference between the velocity of the rain relative to the ground and the velocity of the person relative to the ground. We can use vector subtraction to do this.

Let's define the following vectors:
v_rain_ground = 20 m/s downwards (vector A)
v_person_ground = 5 m/s towards the east (vector B)

We can calculate the velocity of the rain relative to the person as follows:
v_rain_person = v_rain_ground - v_person_ground

Using vector subtraction, we get:
v_rain_person = 20 m/s downwards - 5 m/s towards the east
v_rain_person = 20 m/s downwards + 5 m/s towards the west

Therefore, the velocity of the rain relative to the person is 20 m/s downwards and 5 m/s towards the west.

Step 4: Calculate the angle with the vertical which the person should hold his umbrella so that he may not get drenched.

Let's define the following vectors:
v_rain_person = 20 m/s downwards and 5 m/s towards the west (vector C)

The angle with the vertical which the person should hold his umbrella so that he may not get drenched is the angle between the vector C and the vertical.

We can calculate this angle using trigonometry. Let's define this angle as θ.

Using trigonometry, we get:
tan θ = |C_x|/|C_y|
tan θ = 5/20
tan θ = 0.25

Taking the inverse tangent of both sides, we get:
θ = tan^-1(0.25)

Using a calculator, we get:
θ = 14.04 degrees

Therefore, the angle with the vertical which the person should hold his umbrella so that he may not get drenched is approximately 14.04 degrees.